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A new Hodge operator in discrete exterior calculus. Application to fluid mechanics
Orbital stability and spectral properties of solitary waves of Klein–Gordon equation with concentrated nonlinearity
1. | Texas A & M University, College Station, TX, Institute for Information Transmission Problems, Moscow, Russia |
2. | Vienna University, Vienna, Austria, Institute for Information Transmission Problems, Moscow, Russia |
We obtain explicit characterization of orbital and spectral stability of solitary wave solutions to the $ {\bf{U}}(1) $-invariant 1D Klein–Gordon equation coupled to an anharmonic oscillator. We also give the complete analysis of the spectrum of the linearization at a solitary wave.
References:
[1] |
S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics, 2$^{nd}$ edition, American Mathematical Society, Providence, RI, 2005.
doi: 10.1007/978-3-642-88201-2. |
[2] |
N. Boussaïd and A. Comech, Virtual levels and virtual states of linear operators in Banach spaces. Applications to Schrödinger operators, preprint, arXiv: 2101.11979. |
[3] |
V. Buslaev, A. Komech, E. Kopylova and D. Stuart,
On asymptotic stability of solitary waves in Schrödinger equation coupled to nonlinear oscillator, Commun. Partial Differ. Equ., 33 (2008), 669-705.
doi: 10.1080/03605300801970937. |
[4] |
E. Csobo, F. Genoud, M. Ohta and and J. Royer,
Stability of standing waves for a nonlinear Klein–Gordon equation with delta potentials, J. Differ. Equ., 268 (2019), 353-388.
doi: 10.1016/j.jde.2019.08.015. |
[5] |
A. Comech and D. Pelinovsky,
Purely nonlinear instability of standing waves with minimal energy, Commun. Pure Appl. Math., 56 (2003), 1565-1607.
doi: 10.1002/cpa.10104. |
[6] |
M. Grillakis, J. Shata and W. Strauss,
Stability theory of solitary waves in the presence of symmetry, J. Funct. Anal., 74 (1987), 160-197.
doi: 10.1016/0022-1236(87)90044-9. |
[7] |
A. Jensen and T. Kato,
Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J., 46 (1979), 583-611.
|
[8] |
A. Komech and A. Komech,
Global attractor for a nonlinear oscillator coupled to the Klein–Gordon field, Arch. Ration. Mech. Anal., 185 (2007), 105-142.
doi: 10.1007/s00205-006-0039-z. |
[9] |
A. Komech and A. Komech,
Global attraction to solitary waves for Klein–Gordon equation with mean field interaction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 855-868.
doi: 10.1016/j.anihpc.2008.03.005. |
[10] |
A. Komech, E. Kopylova and D. Stuart,
On asymptotic stability of solitons in a nonlinear Schrödinger equation, Commun. Pure Appl. Anal., 11 (2012), 1063-1079.
doi: 10.3934/cpaa.2012.11.1063. |
[11] |
A. Kolokolov,
Stability of the dominant mode of the nonlinear wave equation in a cubic medium, J. Appl. Mech. Tech. Phys., 14 (1973), 426-428.
doi: 10.1016/0021-8928(74)90131-2. |
[12] |
E. Kopylova,
On the asymptotic stability of solitary waves in the discrete Schrödinger equation coupled to a nonlinear oscillator, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), 3031-3046.
doi: 10.1016/j.na.2009.01.188. |
[13] |
E. Kopylova,
On asymptotic stability of solitary waves in discrete Klein–Gordon equation coupled to a nonlinear oscillator, Appl. Anal., 89 (2010), 1467-1492.
doi: 10.1080/00036810903277176. |
[14] |
M. Murata,
Asymptotic expansions in time for solutions of Schrödinger-type equations, J. Funct. Anal., 49 (1982), 10-56.
doi: 10.1016/0022-1236(82)90084-2. |
[15] |
M. Ohta and G. Todorova,
Strong instability of standing waves for the nonlinear Klein–Gordon equation and the Klein–Gordon–Zakharov system, SIAM J. Math. Anal., 38 (2007), 1912-1931.
doi: 10.1137/050643015. |
show all references
References:
[1] |
S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics, 2$^{nd}$ edition, American Mathematical Society, Providence, RI, 2005.
doi: 10.1007/978-3-642-88201-2. |
[2] |
N. Boussaïd and A. Comech, Virtual levels and virtual states of linear operators in Banach spaces. Applications to Schrödinger operators, preprint, arXiv: 2101.11979. |
[3] |
V. Buslaev, A. Komech, E. Kopylova and D. Stuart,
On asymptotic stability of solitary waves in Schrödinger equation coupled to nonlinear oscillator, Commun. Partial Differ. Equ., 33 (2008), 669-705.
doi: 10.1080/03605300801970937. |
[4] |
E. Csobo, F. Genoud, M. Ohta and and J. Royer,
Stability of standing waves for a nonlinear Klein–Gordon equation with delta potentials, J. Differ. Equ., 268 (2019), 353-388.
doi: 10.1016/j.jde.2019.08.015. |
[5] |
A. Comech and D. Pelinovsky,
Purely nonlinear instability of standing waves with minimal energy, Commun. Pure Appl. Math., 56 (2003), 1565-1607.
doi: 10.1002/cpa.10104. |
[6] |
M. Grillakis, J. Shata and W. Strauss,
Stability theory of solitary waves in the presence of symmetry, J. Funct. Anal., 74 (1987), 160-197.
doi: 10.1016/0022-1236(87)90044-9. |
[7] |
A. Jensen and T. Kato,
Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J., 46 (1979), 583-611.
|
[8] |
A. Komech and A. Komech,
Global attractor for a nonlinear oscillator coupled to the Klein–Gordon field, Arch. Ration. Mech. Anal., 185 (2007), 105-142.
doi: 10.1007/s00205-006-0039-z. |
[9] |
A. Komech and A. Komech,
Global attraction to solitary waves for Klein–Gordon equation with mean field interaction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 855-868.
doi: 10.1016/j.anihpc.2008.03.005. |
[10] |
A. Komech, E. Kopylova and D. Stuart,
On asymptotic stability of solitons in a nonlinear Schrödinger equation, Commun. Pure Appl. Anal., 11 (2012), 1063-1079.
doi: 10.3934/cpaa.2012.11.1063. |
[11] |
A. Kolokolov,
Stability of the dominant mode of the nonlinear wave equation in a cubic medium, J. Appl. Mech. Tech. Phys., 14 (1973), 426-428.
doi: 10.1016/0021-8928(74)90131-2. |
[12] |
E. Kopylova,
On the asymptotic stability of solitary waves in the discrete Schrödinger equation coupled to a nonlinear oscillator, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), 3031-3046.
doi: 10.1016/j.na.2009.01.188. |
[13] |
E. Kopylova,
On asymptotic stability of solitary waves in discrete Klein–Gordon equation coupled to a nonlinear oscillator, Appl. Anal., 89 (2010), 1467-1492.
doi: 10.1080/00036810903277176. |
[14] |
M. Murata,
Asymptotic expansions in time for solutions of Schrödinger-type equations, J. Funct. Anal., 49 (1982), 10-56.
doi: 10.1016/0022-1236(82)90084-2. |
[15] |
M. Ohta and G. Todorova,
Strong instability of standing waves for the nonlinear Klein–Gordon equation and the Klein–Gordon–Zakharov system, SIAM J. Math. Anal., 38 (2007), 1912-1931.
doi: 10.1137/050643015. |


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